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May 10, 2007

Zoo of Lie n-Algebras

Posted by Urs Schreiber

Higher order generalizations of Lie algebras
have equivalently been conceived as
Lie nn-algebras, as L∞L_{\infty}-algebras,
or, dually, as quasi-free differential
graded commutative algebras (quasi-“FDA”s, of “qfDGCA”s).

Here we present a
menagerie of examples concentrated in low degrees,
study their morphisms and discuss applications to
higher order connections, in particular String 2-connections
and Chern-Simons 3-connections.

I’d be grateful for comments.

This is work that has grown out of developments that I talked about in these entries:

30 Comments & 10 Trackbacks

Re: Zoo of Lie n-Algebras

Very cool paper, though often above my level. But what is needed to move from an unstructured “zoo” or “menagerie” to a more complete classification? Is there something like a lattice of such examples? Is there a “height” metric according to which one knows that one has the “smallest” example? What finite objects get their interesting properties from entities in the zoo?

Although L∞L_\infty algebras (or sh Lie algebras) have been objects of much research during the past several years, concrete examples of these structures remain somewhat elusive.

Given that state of the art, the next best thing one should hope to obtain is certainly a good inventory of examples.

And more: we don’t just want arbitrary examples, we want some that actually occur in nature – and we want to gain something for our other tasks by identifying Lie nn-algebras where they have previously remained unidentified (or addressed, irritatingly, as “soft group manifolds”).

Daily and Lada in the above paper end their introduction with the sentence

We leave it as a challenge to the physicists to develop a physical model whose gauge transformations are described by this algebraic examples.

So that’s a solution looking for a problem. We’d rather have it the other way round.

I hope this makes it look more pardonable that the title of our document contains the word “zoo” without it being preceeded by the words “taming of the” (as in the famous Moore-Seiberg paper).

That said, I would like to emphasize that the bulk of the paper goes into considerably more structural investigations than title might suggest.

While lots and lots clearly remains to be better understood, we do amplify a couple of striking patterns that do seem to have significance. Among them clearly is the inn\mathrm{inn}-construction, as discussed in the section “Main Results”.

On the other hand, as mentioned in the section “open problems” there are indications that there is a major structural issue here which we appear to be scratching the surface of – but not more.

Well I am being a little vague here, and that alone is of course part of the answer to your problem: much more needs to be understood, certainly.

But to close with something non-vague, I should at least recall that for semistrict Lie 2-algebras, John Baez and Alissa Crans proved a classification result saying that these are classified by

- a Lie algebra LL

- an LL-module MM

- an element in H3(L,M)H^3(L,M).

This parallels a similar result for Lie 2-groups, given by John Baez and Aaron Lauda.

Re: Zoo of Lie n-Algebras

I should at least recall that for semistrict Lie 2-algebras, John Baez and Alissa Crans proved a classification result saying that these are classified by

- a Lie algebra LL

- an LL-module MM

- an element in the Lie algebra cohomology group H3(L,M)H^3(L,M).

In fact we go a bit further: we classify semistrict Lie nn-algebras CC that are only nontrivial on the bottom and the top — that is, with only the chain groups C0C_0 and Cn−1C_{n-1} nonvanishing. These are classified by:

- A Lie algebra LL

- An LL-module MM

- an element in the Lie algebra cohomology group Hn+1(L,M)H^{n+1}(L,M)

This reduces to the classification you mentioned when n=2n = 2. But, in one small way it’s simpler when nn > 22: in this case we just have L=C0L = C_0 and M=Cn−1M = C_{n-1}.

Of course, one would really like to classify all semistrict Lie nn-algebras. This should also be possible using cohomology — but not the cohomology of Lie algebras. Rather, we should use the cohomology of Lie (n−1)(n-1)-algebras!

The idea is that given a Lie nn-algebra LnL_n, we can form a Lie (n−1)(n-1)-algebra Ln−1L_{n-1} by ‘killing off the nn-chains’. It should be possible to completely describe LnL_n in terms of Ln−1L_{n-1}, a module MnM_n of Ln−1L_{n-1}, and a cohomology class in Hn+1(Ln−1,Mn)H^{n+1}(L_{n-1},M_n).

This recursive analysis of a Lie nn-algebra in terms of ‘layers’ will be familiar to people who know about Postnikov towers. People who don’t can find an explanation in my Lectures on nn-categories and cohomology. See especially the section on ‘cohomology: the layer-cake philosophy’.

Re: Zoo of Lie n-Algebras

While lots and lots clearly remains to be better understood, we do amplify a couple of striking patterns that do seem to have significance. Among them clearly is the inn\mathrm{inn}-construction

I have now included a definition of the general operation
inn(⋅):(Lie−n−algebras)→(Lie−(n+1)−algebras)
\mathrm{inn}(\cdot)
:
(\text{Lie}-n-\text{algebras})
\to
(\text{Lie}-(n+1)-\text{algebras})
in section 3.3.

By itself it’s not really deep, but it clarifies a lot of things, I think.

A nonflat nn-connection with values in the Lie nn-algebra 𝔤(n)\mathfrak{g}_{(n)} is a morphism
dcurv:Vect(X)→inn(𝔤(n)).
d\mathrm{curv} :
\mathrm{Vect}(X)
\to
\mathrm{inn}(\mathfrak{g}_{(n)})
\,.
The general construction applies in particular to the 11-dimensional supergravity 3-form, and hence a field configuration of 11-dimensional supergravity should be a morphism
dcurv:Vect(X)→inn(sugra(10,1)).
d\mathrm{curv} :
\mathrm{Vect}(X)
\to
\mathrm{inn}(\mathrm{sugra}(10,1))
\,.

This is now discussed in a little more detail at the end (but still not sufficiently).

It seems to me that this inn-construciton here automatically enforces what are known as the “rheonomy constraints” in the D’Auria-Fré-like formulation of supergravity, which says that after introducing what they call “Cartan integrable systems” one wants the curvature of the SuGra 3-form to really be a 4-form on spacetime, not something funny.

Re: Zoo of Lie n-Algebras

What […] objects get their interesting properties from entities in the zoo?

Quantum field theories.

One major point is that we present a Lie 3-algebra (7.1.3) which should be the structure Lie 3-algebra governing Chern-Simons theory.

And it is “trivialized” by a string-connection. The inclusion
stringk(𝔤)↪inn(stringk(𝔤))↑∼↑𝔤k↪csk(𝔤)
\array{
\mathrm{string}_k(\mathfrak{g})
&\hookrightarrow&
\mathrm{inn}(\mathrm{string}_k(
\mathfrak{g}))
\\
\uparrow^\sim
&&
\uparrow
\\
\mathfrak{g}_k
&\hookrightarrow&
\mathrm{cs}_k(\mathfrak{g})
}
is, as we show, the Lie nn-algebra version of the statement that a string connection trivializes Chern-Simons theory (section 5.2 in What is an elliptic object?).

(This is, though, I should add, not really explained explicitly in the text, at the moment.)

And that is closely related to 10-dimensional supergravity , the heterotic string and the Green-Schwarz mechanism (see (7.2.6)).

And then there is the Lie 3-algebra which makes 11-dimensional supergravity manifestly a higher gauge theory (7.1.4, but this discussion is, as you can see, not finished yet).

And then, as mentioned in the section “open problems”, there should be a Lie nn-algebra somehow unifying these last three items.

In a word: like to every Lie algebra you can associate a (number of) gauge theory(ies), for every Lie nn-algebra you get corresponding higher gauge theories.

Re: Zoo of Lie n-Algebras

Urs wrote:

What […] objects get their interesting properties from entities in the zoo?

Quantum field theories.

And more generally: anything that has continuous symmetries, symmetries between symmetries, symmmetries between symmetries between symmetries… and so on to the nnth level, will have a Lie nn-algebra associated to it!

Re: Zoo of Lie n-Algebras

Just above proposition 5 you mention INN(G)INN(G) and I just realised that it is the groupoid version of EGEG, namely the realisation of it is Milnor’s universal bundle. This begs the question: what is the corresponding construction for a 2-group? Its Lie 2-algebra will be your construction as the paper. I’m sure I’ve seen you write this before, but I can’t recall where. I can write down EΓE\Gamma for a general groupoid, but I can’t figure out its categorification before I go home tonight. ;D

Re: Zoo of Lie n-Algebras

Just above proposition 5 you mention INN(G)\mathrm{INN}(G) and I just realised that it is the groupoid version of EGE G, namely the realisation of it is Milnor’s universal bundle.

Yes. INN(G)\mathrm{INN}(G) is the strict 2-group coming from the Lie crossed module
id:G→G.
\mathrm{id} : G \to G
\,.
Since that is equivalent to the trivial 2-group, the realization of its nerve is a contractible space, and indeed one finds that
|(G→IdG)|≃EG↓|ΣG|≃BG
\array{
|(G \stackrel{\mathrm{Id}}{\to} G)|
&\simeq& E G
\\
\downarrow
\\
|\Sigma G|
&\simeq&
B G
}
is the universal GG-bundle.

For a general crossed module
(H→G)
(H \to G)
one finds that the realization of the nerve, as a category, of the corresponding 2-group
|(H→G)|
|(H \to G)|
is the classifying space of “HH-bibundles relative to GG” (you know what I mean, everybody has another name for these).

Universal principal 2-bundles

for every group GG, with Disc(G)\mathrm{Disc}(G) the discrete category over the underlying set of GG and with ΣG\Sigma G the 1-object groupoid coming from GG, we have an “exact” sequence of categories
Disc(G)→INN(G)→ΣG
\mathrm{Disc}(G) \to \mathrm{INN}(G)
\to
\Sigma G
(this is one of these points where it really pays to distinguish GG from ΣG\Sigma G).

The important thing is that

a)

INN(G)\mathrm{INN}(G) is equivalent to the trivial 2-group, as one easily checks (using that it’s a codiscrete groupoid).

b)

clearly the preimage of any morphism under the projection
INN(G)→ΣG
\mathrm{INN}(G) \to \Sigma G
is nothing but one copy of GG. In other words: GG acts freely and transitively on the fibers of INN(G)\mathrm{INN}(G) under this projection.

That INN(G)\mathrm{INN}(G) is trivializable translates into EGE G being contractible.

The GG-action translates into the principal GG-action on EGE G.

Now, behind the scenes David Roberts has started working out for simplicial groups what the analog statement of that would be, when we start with the simplicial group coming from a strict 2-group G(2)G_{(2)}.

But without getting into the details of that, it is already easy to check, using the explicit description of INN(G(2))\mathrm{INN}(G_{(2)}) given here, that we do get an exact sequence of 2-categories

Re: Universal principal 2-bundles

For ordinary principal G-bundles,
being universal in the sense of classifying
is equivalent to having total space contractible. Is it known/written somewhere
that the same is true for the universal
G(2)-2-bundle?

Re: Universal principal 2-bundles

For ordinary principal GG-bundles,
being universal in the sense of classifying
is equivalent to having total space contractible. Is it known/written somewhere
that the same is true for the universal
G(2)G_{(2)}-2-bundle?

Hm, good question. I was simply assuming this would be true. But I don’t know!

But maybe if we run through the argument which tells us that EGE G is any contractible space with a principal GG-action, we’ll understand why this must be true for 2-groups, too.

And it’s just as easy to see, actually (once we have the explicit description of INN(G(2))\mathrm{INN}(G_{(2)}), at least, that took some work (straightforward, though) to write down), that the same still holds true for GG replaced by a strict 2-group.

So somehow we want to geometrically realize the nerve of
INN(G(2))→ΣG(2)
\mathrm{INN}(G_{(2)}) \to \Sigma G_{(2)}
and regard that as the universal G(2)G_{(2)}-bundle.

I guess above I already made a mistake in thinking about turning this entirely into a space by taking the Duskin nerve and realizing that.

That might be one way to go (realizing 2-bundles by 1-bundles with the same classification), but maybe here it’s better to realize this only partially, such that we do indeed get a universal 2-bundle (a category, etc.)

It looks like we may have to take the nerve of the 3-group INN(G(2))\mathrm{INN}(G_{(2)}) (a 2-category) ony at top level, such as to retain a 1-category. That would then be our universal 2-bundle.

Oh well. I guess probably in the end David Roberts’ way to proceed here is the best one: do everything in the world of simplicial groups.

Re: Zoo of Lie n-Algebras

David,
You’re quick - or else the current version is more transparent. I struggled for quite a while at the wrong aspects
before recognizing inn as contractible;
in fact, generated by a mapping cone.
Then I finally recalled that for a connected group, inner automorphisms are homtopic to the identity.

As for EG, see also the section on the Weil algebra - which predates Milnor’s construction.

Re: Zoo of Lie n-Algebras

Well I’m trying at present to figure the integrated version of inn(𝔤(n))\mathrm{inn}(\mathfrak{g}_{(n)}), mashing it with Urs’ description of INN(G2)\mathrm{INN}(G_2)here.

Yes, that’s how I found inn(𝔤(2))\mathrm{inn}(\mathfrak{g}_{(2)}) originally (though of course there is a much more immediate way to get it):

the thing is that for Lie nn-groups G(n)G_{(n)} it is very easy to understand what the automorphism Lie (n+1)(n+1)-group AUT(G(n))\mathrm{AUT}(G_{(n)}) shoud be: we simply have

AUT(G(n)):=AutnCat(Σ(G(n)))
\mathrm{AUT}(G_{(n)})
:=
\mathrm{Aut}_{n\mathrm{Cat}}
(\Sigma(G_{(n)}))
(where, as usual, I write ΣG(n)\Sigma G_{(n)} for the nn-group regarded as an nn-groupoid with a single object).

I am saying this, because this is an obvious construction, even though it may be tediuous to actually carry it through in concrete examples.

What is less clear to me is how given just a Lie nn-algebra 𝔤(n)\mathfrak{g}_{(n)} we would construct the Lie (n+1)(n+1)-algebra
DER(𝔤(n))
\mathrm{DER}(\mathfrak{g}_{(n)})
directly, without first integrating everything to nn-groups, taking automorphisms there and then differentiating again.

(I know that Danny Stevenson has been thinkin a lot about DER(⋅)\mathrm{DER}(\cdot), so probably he would know how to do this.)

The same comments apply of course to inner automorphisms. There is an obvious sub Lie (n+1)(n+1)-group
INN(G(n))⊂AUT(G(n))
\mathrm{INN}(G_{(n)})
\subset
\mathrm{AUT}(G_{(n)})
which is that maximal sub (n+1)(n+1)-group whose objects come from conjugating kk-morphisms in G(n)G_{(n)} with objects in G(n)G_{(n)}.

Accordingly, there is then the corresponding Lie (n+1)(n+1)-algebra
inn(𝔤(n)):=Lie(INN(G(n))).
\mathrm{inn}(\mathfrak{g}_{(n)})
:=
\mathrm{Lie}(\mathrm{INN}(G_{(n)}))
\,.

For n=1n=1 this is clear. In that document I tried to compute this for n=2n=2 and G(2)G_{(2)} any strict 2-group.

So, first I computed INN(G(2))\mathrm{INN}(G_{(2)}). Then Lie(INN(G(2)))\mathrm{Lie}(\mathrm{INN}(G_{(2)})).

- and that the equations specifying the differentials of these algebras can be read off directly from the curvature pp-forms of an nn-connection with values in 𝔤(n)\mathfrak{g}_{(n)}.

So what I actually did (around p. 9) was that I computed the pp-form curvatures of a 3-connection with values in the 3-group INN(G(2))\mathrm{INN}(G_{(2)}).

If the strict 2-group G(2)G_{(2)} comes from the crossed module (t:H→G)(t : H \to G), one finds that these connections are given by

A 1-form A∈Ω1(X,Lie(G))A \in \Omega^1(X,\mathrm{Lie}(G)).

A 2-form B∈Ω2(X,Lie(H))B \in \Omega^2(X,\mathrm{Lie}(H)).

A curvature 2-form
β=FA+t*∘B.
\beta = F_A + t_* \circ B
\,.

A curvature 3-form
H=dAB.
H = d_A B
\,.
Satisfying the Bianchi identities
dAβ=t*∘H
d_A \beta = t_* \circ H
and
dAH+β∧B=0.
d_A H + \beta \wedge B = 0
\,.
When one knows how it works (and we describe it in our Zoo) then one can directly read off the qfDGCA from these relations. That qfDGCA is (by canonical equivalence) our Lie 3-algebra inn(𝔤(2))\mathrm{inn}(\mathfrak{g}_{(2)}).

That’s how I computed it originally.

Then, a while ago, I realized that, in the examples where I understand it (n=1,2n = 1,2) there is apparently a simple mechanism which reads in any Lie nn-algebra and spits out a Lie (n+1)(n+1)-algebra which – in the cases that I checked – is the inner derivation Lie (n+1)(n+1)-algebra of the former.

This construction is the ∞\infty-functor
inn(⋅):nLie→(n+1)Lie
\mathrm{inn}(\cdot)
:
n\mathrm{Lie}
\to
(n+1)\mathrm{Lie}
which is described in section 3.3.

This explicit construction at the level of Lie nn-algebras also makes clear why taking inn(⋅)\mathrm{inn}(\cdot) is related to non-vanishing curvatures:

If the Lie nn-algebra 𝔤(n)\mathfrak{g}_{(n)} is built on a graded vector space sVsV, then its inner derivation Lie (n+1)(n+1)-algebra inn(𝔤(n))\mathrm{inn}(\mathfrak{g}_{(n)}) is built on
(sV)⊕(ssV),
(sV) \oplus (ssV)
\,,
where ssVssV is just another copy of sVsV, but with the degree shifted up by one.

One then sees that for every degree of sVsV the connection will send the corresponding curvature to the corresponding degree in ssVssV.

One finds this phenomenon already in ordinary Deligne cohomology, which corresponds to the Lie nn-algebra Lie(ΣnU(1))\mathrm{Lie}(\Sigma^n U(1)): there one has to truncate the complex one is working with one below top level – otherwise all curvatures would have to vanish.

From the general perspective, this is actually a trick that allows the curvature component which would really have to live in ssVssV to be non-vanishing, simply by killing off everything.

This trick only works since Lie(ΣnU(1))\mathrm{Lie}(\Sigma^n U(1)) is nontrivial only in top degree. All the lower degrees give rise to a hierarchy of “fake curvatures” ((p≤n)(p \leq n)-form curvatures). To deal with these – and to allow them to be nonvanishing – we need to make room for them by adding ssVssV to our complexes.

some progress

It seems that we could close two of the “open problems”.

It seems it could be shown that qfDGCAs, their morphisms and derivation homotopies of these in fact form an (∞,1)(\infty,1)-category
ωLie.
\omega \mathrm{Lie}
\,.
Moreover, the construction of the inner derivation Lie (n+1)(n+1)-algebra from a given Lie nn-algebra extends to an ∞\infty functor
inn(⋅):ωLie→ωLie.
\mathrm{inn}(\cdot)
: \omega\mathrm{Lie} \to
\omega \mathrm{Lie}
\,.
This is such that in particular
from
𝔤k≃stringk(𝔤)
\mathfrak{g}_k \simeq
\mathrm{string}_k(\mathfrak{g})
it follows that
inn(𝔤k)≃inn(stringk(𝔤)).
\mathrm{inn}(\mathfrak{g}_k)
\simeq
\mathrm{inn}(
\mathrm{string}_k(\mathfrak{g}))
\,.
With that result in hand it suddenly became clear what to do about the expected but unproven equivalence csk(𝔤)≃inn(stringk(𝔤))\mathrm{cs}_k(\mathfrak{g}) \simeq \mathrm{inn}(\mathrm{string}_k(\mathfrak{g})):

simply check if
inn(𝔤k)≃csk(𝔤).
\mathrm{inn}(\mathfrak{g}_k)
\simeq
\mathrm{cs}_k(\mathfrak{g})
\,.
And that turns out to be an easy computation.

Chern-Simons Lie (2n+1)-algebras

It turns out that there is not just the 1-parameter family of Chern-Simons Lie 3-algebras,
csk(𝔤)
\mathrm{cs}_k(\mathfrak{g})
for any semisimple Lie algebra 𝔤\mathfrak{g}, but there are actually two infinite families of (one-parameter) Chern-Simons Lie (2n+1)(2n+1)-algebras, for all odd nn.

One is defined for 𝔤\mathfrak{g} any matrix Lie algebra. The Lie (2n+1)(2n+1)-algebra
cskn(𝔤)
\mathrm{cs}_k^n(\mathfrak{g})
is characterized by the fact that connections taking values in it ar in bijective correspondence with differential forms
(A,B,C)∈Ω1(X,𝔤)×Ω2n(X)×Ω(2n+1)(X)
(A,B,C)
\in
\Omega^1(X,\mathfrak{g})
\times
\Omega^{2n}(X)
\times
\Omega^{(2n+1)}(X)
such that
C=dB+kCSn(A),
C = d B + k \mathrm{CS}_n(A)
\,,
where CSn(X)\mathrm{CS}_n(X) is the nnth Chern-Simons form of AA.

So the (2n+2)(2n+2)-form curvature of these connections is
dC=kTr((FA)n+1).
d C = k \mathrm{Tr}((F_A)^{n+1})
\,.

The other infinite family is for abelian, but higher form Chern-Simons functionals.

Re: Chern-Simons Lie (2n+1)-algebras

This must be related to what Alissa and I did. We classified all Lie nn-algebras with a Lie algebra LL as objects and a representation VV of LL as (n−1)(n-1)morphisms, in terms of the Lie algebra cohomology group

Hn+1(L,V).H^{n+1}(L,V).

In particular, when L=Lie(G)L = Lie(G) with GG compact connected semisimple, and V=ℝV = \mathbb{R} is the trivial representation, these cohomology groups are just the same as the cohomology groups of the Lie group GG regarded as a space. These cohomology groups are well-known — you can look them up in tables (see page 11 here). So, we get a bunch of Lie nn-algebras.

For example, for G=SU(k)G = SU(k), we have

Hn+1(Lie(G),ℝ)=ℝH^{n+1}(Lie(G), \mathbb{R}) = \mathbb{R}

when n=2,4,6,⋯,2k−2n = 2, 4, 6, \cdots, 2k-2 and

Hn+1(Lie(G),ℝ)=0H^{n+1}(Lie(G), \mathbb{R}) = 0

otherwise. So we get a 1-parameter family of interesting Lie nn-algebras with su(k)su(k) as 0-chains and ℝ\mathbb{R} as (n−1)(n-1)-chains precisely for n=2,4,6,⋯,2k−2n = 2,4,6,\cdots, 2k-2.

I don’t completely understand/remember how you’re defining cskn(g)cs_k^n(g), but since the Lie algebra cocycles I’m talking about are closely related to the Chern-Simons forms you’re describing, I’m sure your Lie 3-algebras, 5-algebras, etc. are closely related to the Lie 2-algebras, 4-algebras etc. that I’m talking about, at least in the case where G=SU(k)G = SU(k) — but in fact more generally. I’m only using SU(k)SU(k) as an example.

But, the interesting stuff happens when we hit Lie algebras that have characteristic classes that are not coming from the formulas you give; then we’ll get other Lie nn-algebras.

Re: Chern-Simons Lie (2n+1)-algebras

Last night, when jogging along the river, I realized that these higher Chern-Simons Lie nn-algebras that I talked about are probably all equivalent to the inns of the Lie algebras that you mention!

Just like what I found for the ordinary Chern-Simons Lie 3-algebra: it’s the inn\mathrm{inn} of the Baez-Crans Lie 2-algebra 𝔤k\mathfrak{g}_k:
csk(𝔤)≃inn(𝔤k).
\mathrm{cs}_k(\mathfrak{g})
\simeq
\mathrm{inn}(\mathfrak{g}_k)
\,.

Re: Chern-Simons Lie (2n+1)-algebras

I will work that out now and write it up.

I think I could check this. Actually an easy computation.

For 𝔤\mathfrak{g} any matrix Lie algebra and an ℝ\mathbb{R}-valued (2n+1)(2n+1)-cocycle on that obtained by forming the trace over the product of 2n+12n+1 elements and then multiplying by a constant kk, let
𝔤kn
\mathfrak{g}_k^n
be the corresponding Lie 2n2n-algebra. Then we have an equivalence (even an isomorphism) of Lie (2n+1)(2n+1)-algebras
inn(𝔤kn)≃cskn(𝔤).
\mathrm{inn}(\mathfrak{g}_k^n)
\simeq
\mathrm{cs}_k^n(\mathfrak{g})
\,.

Clearly, this suggests a more general relation still to be identified. It looks roughly like inn(⋅)\mathrm{inn}(\cdot) takes us from the fourth to the third column in that table on p. 11, which you mentioned, in some sense.

Re: Chern-Simons Lie (2n+1)-algebras

Clearly, this suggests a more general relation still to be identified. It looks roughly like inn(⋅)\mathrm{inn}(\cdot) takes us from the fourth to the third column in that table on p. 11, which you mentioned, in some sense.

Yes, I think I could now get the general statement:

as we know, there is a Baez-Crans Lie nn-algebra
𝔤μ
\mathfrak{g}_\mu
for every (n+1)(n+1)-cocycle μ\mu on gg.

Moreover, for every symmetric invariant polynomial kk of degree (n+1)(n+1) on 𝔤\mathfrak{g} satisfying a certain condition, we get a Lie (2n+1)(2n+1)-algebra
csk(𝔤).
\mathrm{cs}_k(\mathfrak{g})
\,.
This is completely characterized by the fact that a connection taking values in it is a triple (A,B,C)∈Ω1(X,𝔤)×Ω2n(X)×Ω2n+1(X)(A,B,C) \in \Omega^1(X,\mathfrak{g})\times \Omega^{2n}(X)\times \Omega^{2n+1}(X) of differential forms such that
C=dB+CSk(A),
C = d B + \mathrm{CS}_k(A)
\,,
where CSk(A)\mathrm{CS}_k(A) is defined by the invariant polynomial kk by
dCSk(A)=k(FA∧⋯∧FA).
d \mathrm{CS}_k(A) = k(F_A \wedge \cdots \wedge F_A)
\,.

Writing μk\mu_k for the (2n+1)(2n+1)-cocycle which is defined by kk, we have the following equivalence of Lie (2n+1)(2n+1)-algebras (which is even an isomorphism):
inn(𝔤μk)≃csk(𝔤).
\mathrm{inn}(\mathfrak{g}_{\mu_k})
\simeq
\mathrm{cs}_k(\mathfrak{g})
\,.

Re: Chern-Simons Lie (2n+1)-algebras

While the above is right, when you think further about it you’ll realize something is funny here. Alas, I was disrupted by having to sleep a little and now by having to teach a little. Hope to get back here soon with some more clarifications.

Re: Chern-Simons Lie (2n+1)-algebras

Here are some more refinements of the above observations, in particular addressing John’s concern that nothing beyond Baez-Crans type Lie nn-algebras is appearing here.

As was emphasized a lot already, to every Lie algebra n+1n+1-cocycle
μ
\mu
on 𝔤\mathfrak{g}
we obtain a Baez-Crans Lie nn-algebra
𝔤μ
\mathfrak{g}_{\mu}
concentrated in degree 1 and nn.

I am claiming that, in a generalization of that from 𝔤\mathfrak{g} to inn(𝔤)\mathrm{inn}(\mathfrak{g}) we obtain for every degree n+1n+1 symmetric invariant polynomial
k
k
on 𝔤\mathfrak{g} a Lie (2n+1)(2n+1)-algebra
chk(𝔤)
\mathrm{ch}_k(\mathfrak{g})
concentrated in degree 1,2 and 2n+12n+1.

I believe that these are not equivalent, in general, to any 𝔤μ\mathfrak{g}_\mu.

I call these chk(𝔤)\mathrm{ch}_k(\mathfrak{g}) for Chern. They are like the Chern-Simons Lie (2n+1)(2n+1)-algebras that I kept talking about, only that the latter has more entries in more degrees.

The Chern-Simons Lie 2n+12n+1-algebra exists when the 2n+12n+1 cochain μk\mu_k defined by the inavriant polynomial kk is a cocycle, and when kk is trivial with respect to the cohomology of inn(𝔤)\mathrm{inn}(\mathfrak{g}) (which doesn’t mean that it is trivial in the ordinary sense). (All this is described in more detail in section 10 now, cleaner than what I had before, I think ).

The Chern-Simons Lie (2n+1)(2n+1)-algebra
csk(𝔤)
\mathrm{cs}_k(\mathfrak{g})
is isomorphic (even) to the inner derivation Lie (2n+1)(2n+1)-algebra of the corresponding Baez-Crans Lie 2n2n-algebra
inn(𝔤μk)≃csk(𝔤).
\mathrm{inn}(\mathfrak{g}_{\mu_k})
\simeq
\mathrm{cs}_k(\mathfrak{g})
\,.
There is now a general proof included (section
3.3) that
inn(𝔤(n))
\mathrm{inn}(\mathfrak{g}_{(n)})
is always trivializable., for all Lie nn-algebras 𝔤(n)\mathfrak{g}_{(n)}.

So this means that the Chern-Simons Lie (2n+1)(2n+1)-algebra always is trivializable as well! – which is why I said above that something funny is going on.

But it all makes sense:

we have a canonical epimorphism
csk(𝔤)→chk(𝔤)
\mathrm{cs}_k(\mathfrak{g})
\to
\mathrm{ch}_k(\mathfrak{g})
which extracts the nontrivial information, namely the Chern-class coming from kk – in the sense that when you consider connections into these beasts, this epimorphism extracts their (2n+2)(2n+2)-form curvature
dC=k((FA)n+1).
d C = k((F_A)^{n+1})
\,.

where the left side is the cohomology of the space GG, but the right side is Lie algebra cohomology.

On the other hand, we have:

Theorem: The cohomology ring

H*(G,ℝ)≅H*(g,ℝ)H^*(G, \mathbb{R}) \cong H^*(g, \mathbb{R})

is isomorphic to an exterior algebra on the elements trans(xi)trans(x_i), which are of odd degree.

On the one hand, according to what you say, each element of H2n(BG,ℝ)H^{2n}(B G, \mathbb{R}) gives a ‘Chern’ Lie (2n−1)(2n-1)-algebra.

On the other hand, each element of H2n−1(G,ℝ)H^{2n-1}(G,\mathbb{R}) gives a ‘Baez–Crans’ Lie (2n−2)(2n-2)-algebra.

In particular, each element xix_i of degree nn spawns a Lie (2n−1)(2n-1)-algebra — but also, via its transgression trans(xi)trans(x_i), a Lie (2n−2)(2n-2)-algebra!

I hope I have the numbers right here. The ‘classic’ example, the one we know best, is supposed to be n=2n = 2. Here we get a Baez–Crans Lie 2-algebra and a Chern Lie 3-algebra. In this case, the relevant invariant homogeneous polynomial on gg has degree 2. It’s often called the 2nd Chern class:

since the cohomology of BGB G is a polynomial algebra, while the cohomology of GG is an exterior algebra.

So, unless I’m confused, there are a lot more interesting ‘Chern’ Lie (2n−1)(2n-1)-algebras than ‘Baez–Crans’ ones.

I think the only elements transgression doesn’t kill are linear combinations of the generators xix_i. For SU(k)SU(k) these generators are just the Chern classes c1,c2,…,ckc_1, c_2, \dots, c_k. For other simple Lie algebras, you can see the degrees of the generators in that table on page 11 here.

Needless to say, Stasheff knows a lot about this stuff: together with Milnor, he wrote the book on characteristic classes!

Re: Zoo of Lie n-Algebras

WARNING! WARNING! WARNING! That’s Chern’s use of transgression; most of the literature on characteristic classes (especially Borel-Serre) calls that suspension and transgression goes the other way - from a subspace of the cohomology of G (aka the transgressive elements) to the cohomology of BG - any coefficients.

So my comments below are in the Borel-Serre language.

Except for the particulars about compact Lie groups (the relation of the cohomology of G to that of its Lie algebra) transgression and suspension work for any sha-space, e.g. the based loops space ΩX for which BΩX is homotopy equivalent to X

Suspension kills not only all products but all Massey products etc

In the special case John mentions, the transgressive elements are precisely the primitive elements - but this is not true in general - cf. my birth certificate.

As for Chern Lie (2n-1)-algebras, consider perhaps more provocatively the Chern-Simons forms which are closed only on manifolds for which they are top dimensional. Indeed, they are the forms that mediate transgression - whichever way you think it is going.

See also the Weil algebra section of our Zoo where there is a revisionist description of the Weil algebra in terms of a Lie 2-algebra and how it played the role of forms on the universal G-bundle before there was one.

In the nice case John describes, there is an analogous cohomology version:

H(G) ⊗ H(BG) with a twisted differential making it acyclic, i.e. the Koszul complex which in this case is a resolution of the ground field.

Re: Zoo of Lie n-Algebras

See also the Weil algebra section of our Zoo where there is a revisionist description of the Weil algebra in terms of a Lie 2-algebra and how it played the role of forms on the universal GG-bundle before there was one.

The Lie algebra n+1n+1 cocycles are precisely the elements in
Hn+1(dg).
H^{n+1}(d_g)
\,.
This should correspond to the second theorem that John mentioned.

Then, the symmetric invariant polynomials of degree n+1n+1 are precisely the elements in
H2n+2(dinn(g))|(ssg)*.
H^{2n+2}(d_{\mathrm{inn}(g)})|_{(s s g)^*}
\,.
That should essentially be the Chern-Weil theorem that John mentioned at the beginning of his comment.

Now, how does that story continue?

What is the good way to say that a given invariant polynomial admits a Chern-Simons form?

I think the point is that the polynomial kk must be cohomologically nontrivial (with respect to dinn(g)d_{\mathrm{inn}(g)}) as an element in ∧•(ssg)*\wedge^\bullet (s s g)^*, but not as an element of
∧•((sg)*⊕(ssg)*)\wedge^\bullet ( (s g)^* \oplus (s s g)^* ).

As an element of the latter it trivializes
k(r)=dinn(g)(f(t,r))
k(r) = d_{\mathrm{inn}(g)}(f(t,r))
and the potential ff gives the Chern-Simons form – at least when ff looks like
f(t,r)=μk(t)+Q(t,r),
f(t,r) = \mu_k(t) + Q(t,r)
\,,
where μk(t)\mu_k(t) is the (2n+1)(2n+1)-cocycle coming from the degree (n+1)(n+1) polynomial kk, and where Q(t,r)Q(t,r) is some polynomial which is at least linear in rr (i.e. which vanishes when restricted to ∧•(sg)*\wedge^\bullet (s g)^*).

So, what is the more sophisticated way of saying all this? What’s really going on when a Chern-Simons form exists for a given invariant polynomial? Does it say anything beyond that μk\mu_k is a cocycle, hence that we have transgression here relating invariant polynomial and cocycle?

Since inn(g)\mathrm{inn}(g) trivializes as a Lie 2-algebra, this should mean that the dinn(g)d_{\mathrm{inn}(g)}-cohomology is trivial (but I am confused slightly about this point, related to that problem with derivation homotopies).

But if this is true, then it would mean that every invariant polynomial is cohomologically trivial when extended from ∧•(ssg)*\wedge^\bullet (s s g)^*, where it is born, to all of ∧•((sg)*⊕(ssg)*)\wedge^\bullet( (s g)^* \oplus (s s g)^* ).

So then it would seem that the only further condition for a Chern-Simons form to exist is that a QQ exists, as above.

What does that really mean? Is that even an extra condition? What is going on here?

Whatever the answer is, it would greatly help me to see it entirely in the language of inn(g)\mathrm{inn}(g), if possible.

Re: Zoo of Lie n-Algebras

Transgression is what it is all about.
Here’s a way to describe transgression
(Cartan-Serre-Borel terminology.

H here will be cohomology p
Consider a fibration F –> E –> B
and a form w representing a class in H(F).
Extend w to a form v on all of E.
dv may be non-zero but dv = p^* u
where u is a closed form on B
then the class of u is the transgression of the class of w.

Fortunately in the case of Chern-Weil and the Weil algebra, this is all very specific in terms of polynomials -
exterior/Grassmann for G and polynomial for BG

There are various standard bases for invariant symmetric polynomials on g;
these represent bases for the characteristic classes of G bundles.

For u = Tr(F^n)
v can be taken to be
Tr(A F^n-1)

This is the Chern-Simons form - NOT
closed in general
BUT if B is of dim 2n-1,
then it is closed and basic and carries
geometric info - which is what Chern and Simons were after.

Re: Zoo of Lie n-Algebras

Consider a fibration F→E→BF \to E \to B
and a form ww representing a class in H(F)H(F).
Extend WW to a form vv on all of EE.
dvd v may be non-zero but dv=p*ud v = p^* u
where uu is a closed form on BB
then the class of u is the transgression of the class of ww.

Thanks. I think this I understand. For instance, concretely, where p:E→Bp : E \to B is a principal GG-bundle (probably for GG
compact simple and simply connected) with connection AA, then
⟨FA∧FA⟩
\langle F_A \wedge F_A \rangle
is a closed 4-form on BB and its pullback p*⟨FA∧FA⟩p^* \langle F_A \wedge F_A \rangle to EE is exact
p*⟨FA∧FA⟩=⟨A∧FA⟩−16⟨A∧[A∧A]⟩
p^* \langle F_A \wedge F_A \rangle
=
\langle A \wedge F_A\rangle
-
\frac{1}{6}\langle A \wedge [A\wedge A]\rangle
and the right hand side restricts to
−16⟨θ∧[θ∧θ]⟩
-
\frac{1}{6}\langle
\theta \wedge [\theta\wedge \theta]\rangle
on the fibers of EE.

Fortunately in the case of Chern-Weil and the Weil algebra, this is all very specific in terms of polynomials -
exterior/Grassmann for GG and polynomial for BGB G

And this I need to understand in terms of the qfDGCA of inn(g)\mathrm{inn}(g). I’ll take a look at your book on Monday.

It’s these further terms (their coefficients especially) that I find a little hard to come by.

I am thinking that there is a degree -1 map τ\tau on the qfDGCA of inn(g)\mathrm{inn}(g) which satisfies
[dinn(g),τ]=Idinn(g).
[d_{\mathrm{inn}(g)},\tau]
=
\mathrm{Id}_{\mathrm{inn}(g)}
\,.
If so, then all these terms should be computed as
τ(k(r))
\tau(k(r))
I think.

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